If X is invested in a bank at a rate of simple interest of y% p.a. for

My approach of solving this -

Y1 interest will be same for CI as well as SI i.e. 400
Y2 the extra 20 is what we are earning from CI, which is nothing but SI on interest earned in 1st year
i.e. (400 * Y * 1)/100 = 20
From here we get Y = 5%
And hence X -> (8000 by X * 5 *2 )/100 = 800 gives X= 8000

Solution:

Using the simple interest formula (Principal x rate x time = Interest) we have:

X * y/100 * 2 = 800

Xy/50 = 800

y = 40,000/X

Using the compound interest formula [P(I + r/n)^rt - P = I, where P = principal, r = annual interest rate, n = the number of compounding periods per year, t = time in years, and I = the total interest earned, we have:

X(1 + y/100)^2 - X = 820

X(1 + 2y/100 + y^2/10,000 - 1) = 820

Since y = 40,000/X, we have:

X(2(40,000/X)/100 + (40,000/X)^2/10,000) = 820

X(800/X + 160,000/X^2) = 820

800 + 160,000/X = 820

160,000/X = 20

X = 8,000

Alternate Solution:

The question mentions no units, so we will use dollars for clarity.

Since the interest is simple, exactly half of the interest of $800 is the annual interest; in other words, the principal of X acquires an interest of 800/2 = $400 in one year.

When the interest is compounded annually, we see that 820 - 800 = $20 more interest is earned. This is because in the second term, the principal acquiring interest is X + 400 dollars instead of X dollars. Thus, the extra interest of $20 is y percent of $400. Let’s use this to calculate y:

400 * y/100 = 20

y = 5

Now that we know y = 5, let’s use the fact that X dollars generate $400 in interest in one year at 5 percent:

X * 5/100 = 400

X = $8,000

Answer: A

If X is invested in a bank at a rate of simple interest of y% p.a. for two years, then the interest earned is 800. if X is invested at y% p.a., for two years when the interest is compounded annually, the interest is 820. What is the value of X?

A. 8000
B. 6000
C. 5000
D. 4000
E. 3000

1. Simple interest is the exact same amount for all the years
2. Compound interest = Simple interest (for first year)
3. Compound interest - Simple interest for (first two years) = Interest on the interest for the 1st year

Now, coming back to the problem in hand
If SI is $800 for 2 years, the simple interest for one year is \($800/2\) = $400

Compound interest(2nd year) = $400*2 + Interest on $400 = $20 (820-800)
We can calculate the rate of interest as \(20/400\)* 100 = 5%. Now, as $400 is the interest for the first year and we have found the rate of interest to be 5%, the principal is \(400/0.05\) = $8000

A